
i) \(\Omega (z) = \frac{1}{p}\exp (z^p)z^{2-p}\)  for \(0<p\le 1\)

ii) \(\Omega (z) = \frac{\pi }{2}(\exp (2z) - 1)^{1/2}z\)

iii) \(\Omega (z) = \gamma z(a z^p+b z^q)^k\)  where \(a,b,p,q,k>0\)  satisfy \(kp<1\)  and \(kq>1\) , and
\(\gamma =\frac{\Gamma \left(m_p\right) \Gamma \left(m_q\right)}{a^{k}\Gamma (k) (q-p)}\left(\frac{a}{b}\right)^{m_p},\) 
with \(m_p=\frac{1-kp}{q-p}\) , \(m_q=\frac{kq-1}{q-p}\)  and \(\Gamma (\cdot )\)  is the Gamma function defined as \(\Gamma (z)=\int _0^{+\infty } e^{-t}t^{z-1}dt\)  [1]}.


